Polytope Research Articles

Orientifold Calabi-Yau threefolds: divisor exchanges and multi-reflections

Using the Kreuzer-Skarke database of 4-dimensional reflexive polytopes, we systematically constructed a new database of orientifold Calabi-Yau threefolds with h1,1(X) ≤ 12. Our approach involved non-trivial ℤ2 involutions, incorporating both divisor exchanges and multi-divisor reflections acting on the Calabi-Yau threefolds. Each proper involution results in an orientifold Calabi-Yau threefolds and we constructed 320, 386, 067 such examples. We developed a novel algorithm that significantly reduces the complexity of determining all the fixed loci under the involutions, and clarifies the types of O-planes. Our results show that under proper involutions, the majority of cases end up with O3/O7-plane systems, and most of these further admit a naive Type IIB string vacua. Additionally, a new type of free action was determined. We also computed the smoothness and the splitting of Hodge numbers in the ℤ2-orbifold limit for these orientifold Calabi-Yau threefolds.

Integer Decomposition Property of Polytopes

We study the integer decomposition property of lattice polytopes associated with the $n$-dimensional smooth complete fans with at most $n+3$ rays. Using the classification of smooth complete fans by Kleinschmidt and Batyrev and a reduction to lower dimensional polytopes we prove the integer decomposition property for lattice polytopes in this setting.

Recursive constructions for the higher Stasheff—Tamari orders in dimension three using the Outer Tamari and Tamari Block posets

We define a poset called the Outer Tamari poset, and show it is isomorphic to a subposet of the Tamari lattice introduced by Pallo (1986), and studied further and called the Comb poset by Csar, Sengupta, and Suksompong (2014). We use the Outer Tamari poset to develop recursive formulas for the number of triangulations of the 3-dimensional cyclic polytopes. These triangulations can be viewed as elements of both the higher Stasheff–Tamari orders in dimension three and the Tamari Block lattices we defined in a previous article. So our work here can also be seen as constructing recursive enumerations of these posets.

Deep Lattice Points in Zonotopes, Lonely Runners, and Lonely Rabbits

Abstract Let $P \subseteq {\mathbb {R}}^{d}$ be a polytope and let $\textbf {w}$ be an interior point of $P$. The coefficient of asymmetry$\operatorname {ca}(P,\textbf {w}):= \min \{ \lambda \geq 1: \textbf {w} - P \subseteq \lambda (P - \textbf {w}) \}$ of $P$ about $\textbf {w}$ has been studied extensively in the realm of Hensley’s conjecture on the maximal volume of a $d$-dimensional lattice polytope that contains a fixed positive number of interior lattice points. We zero in on the coefficient of asymmetry for lattice zonotopes, that is, Minkowski sums of line segments with integer endpoints. Our main result gives the existence of an interior lattice point for which the coefficient of asymmetry is bounded above by an explicit constant in $\Theta (d \log \log d)$, for any lattice zonotope that has an interior lattice point. Our work is both inspired by and feeds on Wills’ lonely runner conjecture from Diophantine approximation: we make intensive use of a discrete version of this conjecture (which, in fact, has been proved), and reciprocally, we reformulate the lonely runner conjecture in terms of the coefficient of asymmetry for certain lattice zonotopes.

Computations of volumes in five candidates elections

We describe several analytical (i.e., precise) results obtained in five candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality, negative plurality and Borda voting, including their runoff versions. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes in dimension 119, using its recent implementation of the Lawrence algorithm.

A node elimination algorithm for cubature of high-dimensional polytopes

Node elimination is a numerical approach for obtaining cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. This paper introduces a new criterion for selecting which nodes to eliminate that is based on a linearization of the moment equation. In addition, a penalized iterative solver is introduced, that ensures that weights are positive and nodes are inside the integration domain. A strategy for constructing an initial cubature rule for various polytopes in several space dimensions is described. High efficiency rules are presented for two, three and four dimensional polytopes. The new rules are compared with rules that are obtained by combining tensor products of one dimensional quadrature rules and domain transformations, as well as with known analytically constructed cubature rules.

The Positive Tropical Grassmannian, the Hypersimplex, and them= 2 Amplituhedron

AbstractThe positive Grassmannian $Gr^{\geq 0}_{k,n}$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map$\mu $ onto the hypersimplex [ 31] and the amplituhedron map$\tilde{Z}$ onto the amplituhedron [ 6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills. We define a map we call T-duality from cells of $Gr^{\geq 0}_{k+1,n}$ to cells of $Gr^{\geq 0}_{k,n}$ and conjecture that it induces a bijection from positroid dissections of the hypersimplex $\Delta _{k+1,n}$ to positroid dissections of the amplituhedron $\mathcal{A}_{n,k,2}$; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an $(n-1)$-dimensional polytope while the amplituhedron $\mathcal{A}_{n,k,2}$ is a $2k$-dimensional non-polytopal subset of the Grassmannian $Gr_{k,k+2}$. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher $m$, we define the momentum amplituhedron for any even $m$.

Fusions of Consciousness.

What are conscious experiences? Can they combine to form new experiences? What are conscious subjects? Can they combine to form new subjects? Most attempts to answer these questions assume that spacetime, and some of its particles, are fundamental. However, physicists tell us that spacetime cannot be fundamental. Spacetime, they say, is doomed. We heed the physicists, and drop the assumption that spacetime is fundamental. We assume instead that subjects and experiences are entities beyond spacetime, not within spacetime. We make this precise in a mathematical theory of conscious agents, whose dynamics are described by Markov chains. We show how (1) agents combine into more complex agents, (2) agents fuse into simpler agents, and (3) qualia fuse to create new qualia. The possible dynamics of n agents form an n(n-1)-dimensional polytope with nn vertices-the Markov polytopeMn. The total fusions of n agents and qualia form an (n-1)-dimensional simplex-the fusion simplexFn. To project the Markovian dynamics of conscious agents onto scattering processes in spacetime, we define a new map from Markov chains to decorated permutations. Such permutations-along with helicities, or masses and spins-invariantly encode all physical information used to compute scattering amplitudes. We propose that spacetime and scattering processes are a data structure that codes for interactions of conscious agents: a particle in spacetime is a projection of the Markovian dynamics of a communicating class of conscious agents.

Some notes on generic rectangulations

A rectangulation is a subdivision of a rectangle into rectangles. A generic rectangulation is a rectangulation that has no crossing segments. We explain several observations and pose some questions about generic rectangulations. In particular, we show how one may "centrally invert" a generic rectangulation about any given rectangle, analogous to reflection across a circle in classical geometry. We also explore 3-dimensional orthogonal polytopes related to "marked" rectangulations and drawings of planar maps. These observations arise from viewing a generic rectangulation as topologically equivalent to a sphere.

Complex psd-minimal polytopes in dimensions two and three

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last of these, for which the least is known, and in particular on understanding which polytopes are complex psd-minimal. We prove the existence of an obstruction to complex psd-minimality which is efficiently computable via lattice membership problems. Using this tool, we complete the classification of complex psd-minimal polygons (geometrically as well as combinatorially). In dimension three we exhibit several new examples of complex psd-minimal polytopes and apply our obstruction to rule out many others.

Brill-Noether-general limit root bundles: absence of vector-like exotics in F-theory Standard Models

Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations (3,2)1/6, ( overline{textbf{3}} ,1)−2/3 and (1,1)1. For the family B3( {Delta }_4^{{}^{circ}} ) consisting of mathcal{O} (1011) F-theory QSM geometries, we argue that more than 99.995% of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario in the mathcal{O} (1011) QSM geometries B3( {Delta }_4^{{}^{circ}} ).The QSM geometries come in families of toric 3-folds B3(∆°) obtained from triangulations of certain 3-dimensional polytopes ∆°. The matter curves in XΣ ∈ B3(∆°) can be deformed to nodal curves which are the same for all spaces in B3(∆°). Therefore, one can probe the vector-like spectra on the entire family B3(∆°) from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves.In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these jumping circuits, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. B3( {Delta }_4^{{}^{circ}} ) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.

The edge-transitive polytopes that are not vertex-transitive

In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show that these polyhedra do not have higher-dimensional analogues, that is, that in dimension d ≥ 4, edge-transitivity of convex polytopes implies vertex-transitivity.More generally, we give a classification of all convex polytopes which at the same time have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show that any such polytope in dimension d ≥ 4 is vertex-transitive.

A reduced LPV polytopic look-ahead steering controller for autonomous vehicles

This paper presents a novel design of a Linear Parameter Varying (LPV) controller based-on the polytopic approach, for the path-following system of an automated vehicle. The implementation of the proposed steering system is based-on the reduction of the initial 3D (3 Dimensional) polytope that, due to the conservatism of the specific approach suffers from over-bounding. The proposed algorithm aims at tightening the polytope by reducing the vertices and the volume of the polytope that includes the parameter variations i.e the longitudinal velocity of the car, its inverse and, the look-ahead distance of the steering algorithm. In this paper, a novel real-time implementation of the reduced LPV/Polytopic controller is proposed as a real-time optimization problem according to the current measured values of parameters. The novel control system, has been validated in high-fidelity simulations, and to a real-life experiment at a test-bed platform (Renault Zoe) in a private test track for low and higher speeds showing encouraging performance and sustaining good tracking and comfort at the same time.

Low dimensional flow polytopes and their toric ideals

The toric ideal of a d-dimensional flow polytope has an initial ideal generated by square-free monomials of degree at most d. The toric ideal of a flow polytope of dimension at most four has an initial ideal generated by square-free monomials of degree at most two, with the only exception of the four-dimensional Birkhoff polytope, whose toric ideal has an initial ideal generated by a square-free cubic monomial. The proof is based on a method to classify certain compressed flow polytopes, and a construction of a quadratic pulling triangulation of them. Along the way compressed flow polytopes are classified up to dimension four, and their Ehrhart polynomials are computed.

Linkedness of Cartesian products of complete graphs

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2 k vertices is k -linked if, for every set of 2 k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product K d 1 +1 × K d 2 +1 of complete graphs K d 1 +1 and K d 2 +1 is ⌊( d 1 + d 2 )/2⌋ -linked for d 1 , d 2 ≥ 2 , and this is best possible. This result is connected to graphs of simple polytopes. The Cartesian product K d 1 +1 × K d 2 +1 is the graph of the Cartesian product T ( d 1 ) × T ( d 2 ) of a d 1 -dimensional simplex T ( d 1 ) and a d 2 -dimensional simplex T ( d 2 ) . And the polytope T ( d 1 ) × T ( d 2 ) is a simple polytope , a ( d 1 + d 2 ) -dimensional polytope in which every vertex is incident to exactly d 1 + d 2 edges. While not every d -polytope is ⌊ d /2⌋ -linked, it may be conjectured that every simple d -polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.

Algebraic Degrees of 3-Dimensional Polytopes

Results of Koebe (Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl. 88, 141–164, 1936), Schramm (Invent. Math. 107(3), 543560, 1992), and Springborn (Math. Z. 249, 513–517, 2005) yield realizations of 3-polytopes with edges tangent to the unit sphere. Here we study the algebraic degrees of such realizations. This initiates the research on constrained realization spaces of polytopes.

Extension complexity of low-dimensional polytopes

Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P P is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which P P can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension d d , the extension complexity of a random d d -dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic n n -vertex polygon (whose vertices lie on a circle) has extension complexity at most 24 n 24\sqrt n . This bound is tight up to the constant factor 24 24 . Finally, we show that there exists an n o ( 1 ) n^{o(1)} -dimensional polytope with at most n n vertices and extension complexity n 1 − o ( 1 ) n^{1-o(1)} . Our theorems are proved with a range of different techniques, which we hope will be of further interest.

Improved polytope volume calculations based on Hamiltonian Monte Carlo with boundary reflections and sweet arithmetics

Computing the volume of a high dimensional polytope is a fundamental problem in geometry, also connected to the calculation of densities of states in statistical physics, and a central building block of such algorithms is the method used to sample a target probability distribution. This paper studies Hamiltonian Monte Carlo (HMC) with reflections on the boundary of a domain, providing an enhanced alternative to Hit-and-run (HAR) to sample a target distribution restricted to the polytope. We make three contributions. First, we provide a convergence bound, paving the way to more precise mixing time analysis. Second, we present a robust implementation based on multi-precision arithmetic, a mandatory ingredient to guarantee exact predicates and robust constructions. We however allow controlled failures to happen, introducing the Sweeten Exact Geometric Computing (SEGC) paradigm. Third, we use our HMC random walk to perform H-polytope volume calculations, using it as an alternative to HAR within the volume algorithm by Cousins and Vempala. The systematic tests conducted up to dimension $n$ = 100 on the cube, the isotropic and the standard simplex show that HMC significantly outperforms HAR both in terms of accuracy and running time. Additional tests show that calculations may be handled up to dimension $n$ = 500. These tests also establish that multiprecision is mandatory to avoid exits from the polytope.

Statistics of limit root bundles relevant for exact matter spectra of F-theory MSSMs

In the largest, currently known, class of one Quadrillion globally consistent F-theory Standard Models with gauge coupling unification and no chiral exotics, the vector-like spectra are counted by cohomologies of root bundles. In this work, we apply a previously proposed method to identify toric base 3-folds, which are promising to establish F-theory Standard Models with exactly three quark-doublets and no vector-like exotics in this representation. The base spaces in question are obtained from triangulations of 708 polytopes. By studying root bundles on the quark doublet curve $C_{(\mathbf{3},\mathbf{2})_{1/6}}$ and employing well-known results about desingularizations of toric K3-surfaces, we derive a \emph{triangulation independent lower bound} $\check{N}_P^{(3)}$ for the number $N_P^{(3)}$ of root bundles on $C_{(\mathbf{3},\mathbf{2})_{1/6}}$ with exactly three sections. The ratio $\check{N}_P^{(3)} / N_P$, where $N_P$ is the total number of roots on $C_{(\mathbf{3},\mathbf{2})_{1/6}}$, is largest for base spaces associated with triangulations of the 8-th 3-dimensional polytope $\Delta^\circ_8$ in the Kreuzer-Skarke list. For each of these $\mathcal{O}( 10^{15} )$ 3-folds, we expect that many root bundles on $C_{(\mathbf{3},\mathbf{2})_{1/6}}$ are induced from F-theory gauge potentials and that at least every 3000th root on $C_{(\mathbf{3},\mathbf{2})_{1/6}}$ has exactly three global sections and thus no exotic vector-like quark-doublet modes.

Violation of all two-party facet Bell inequalities by almost-quantum correlations

The characterization of the set of quantum correlations is a problem of fundamental importance in quantum information. The question whether every proper (tight) Bell inequality is violated in quantum theory is an intriguing one in this regard. Here we make significant progress in answering this question, by showing that every tight Bell inequality is violated by ``almost-quantum'' correlations, a semidefinite programming relaxation of the set of quantum correlations. As a consequence, we show that many (classes of) Bell inequalities including two-party correlation Bell inequalities and multioutcome nonlocal computation games, which do not admit quantum violations, are not facets of the classical Bell polytope. To do this, we make use of the intriguing connections between Bell correlations and the graph-theoretic Lov\'asz-theta set, discovered by Cabello-Severini-Winter (CSW). We also exploit connections between the cut polytope of graph theory and the classical correlation Bell polytope, to show that correlation Bell inequalities that define facets of the lower dimensional correlation polytope are violated in quantum theory.